A126063 Triangle read by rows: see A128196 for definition.

1, 1, 2, 3, 6, 4, 15, 30, 20, 8, 105, 210, 140, 56, 16, 945, 1890, 1260, 504, 144, 32, 10395, 20790, 13860, 5544, 1584, 352, 64, 135135, 270270, 180180, 72072, 20592, 4576, 832, 128, 2027025, 4054050, 2702700, 1081080, 308880, 68640, 12480, 1920, 256

Offset: 0

Author

N. J. A. Sloane, Feb 28 2007

Keywords

• nonn
• tabl
• easy

Examples

			Triangle begins:
       1
       1,       2
       3,       6,       4
      15,      30,      20,       8
     105,     210,     140,      56,     16
     945,    1890,    1260,     504,    144,    32
   10395,   20790,   13860,    5544,   1584,   352,    64
  135135,  270270,  180180,   72072,  20592,  4576,   832,  128
		

Links

• Ivan Neretin, Rows n = 0..100, flattened
• P. Luschny, Variants of Variations.

Crossrefs

First column is A001147, second column is A097801.

The diagonal is A000079, the subdiagonal is A014480.

Programs

Maple

A126063 := (n,k) -> 2^k*doublefactorial(2*n-1)/ doublefactorial(2*k-1); seq(print(seq(A126063(n,k),k=0..n)),n=0..7); # Peter Luschny, Dec 20 2012

Mathematica

Flatten[Table[2^k (2n - 1)!!/(2k - 1)!!, {n, 0, 8}, {k, 0, n}]] (* Ivan Neretin, May 11 2015 *)

Formula

Let H be the diagonal matrix diag(1,2,4,8,...) and

let G be the matrix (n!! defined as A001147(n), -1!! = 1):

(-1)!!/(-1)!!

1!!/(-1)!! 1!!/1!!

3!!/(-1)!! 3!!/1!! 3!!/3!!

5!!/(-1)!! 5!!/1!! 5!!/3!! 5!!/5!!

...

Then T = G*H. [Gottfried Helms]

T(n,k) = 2^k*(2n - 1)!!/(2k - 1)!!. - Ivan Neretin, May 13 2015